A digit is a number in the set{0, 1, . . . , 9}. A letter is an element of the s

Question

A digit is a number in the set{0, 1, . . . , 9}. A letter is an element of the set {a,b, . . . ,z} {A,B, . . . ,Z}; the first set is the set of lowercase letters and the second is the set of uppercase letters. A special character is an element of the set {#, $, %, &,}.

On Dave’s computer system, a valid password is a sequence of 8 to 12 symbols, where each symbol is a digit or a letter or a special character and there is at least one special character and at least one uppercase letter. Count the number of possible valid passwords on Dave’s system

Explanation / Answer

8 to 12 symbols

There are 10 numerals, 26 lower case letters, 26 uppercase letters and 5 special characters.

So there are (10+26+26+5) = 67 ways to choose one symbol.

So there are (67)8 ways to form an 8 character long password.

The non-eligible ones are the ones which do not have at least one special character and there are (62)8 of them.

Also which do not have at least one uppercase and there are (41)8 of them.

So the total number of eligible 8 character long passwords are

= (67)8  - (  (62)8 + (41)8 - (36)8) [  (36)8 is the intersection of the two sets ]

In the same way, we can compute the number of eligible passwords of 9,10,11,12 character lengths.

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